2log (problem 3.3.6)

Average Error: 29.5 → 0.0

Time: 5.3min

Precision: binary64

• Falling back on racket egraph because egg-herbie package not installed

Math

\[\log \left(N + 1\right) - \log N\]

↓

\[\begin{array}{l} \mathbf{if}\;N \le 7318.37632317876796:\\ \;\;\;\;\log \left(\sqrt{\frac{N + 1}{N}}\right) + \log \left(\sqrt{\frac{N + 1}{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} \cdot \left(1 + \frac{\frac{0.333333333333333315}{N} - 0.5}{N}\right)\\ \end{array}\]

C

double code(double N) {
	return ((double) (((double) log(((double) (N + 1.0)))) - ((double) log(N))));
}

↓

double code(double N) {
	double VAR;
	if ((N <= 7318.376323178768)) {
		VAR = ((double) (((double) log(((double) sqrt((((double) (N + 1.0)) / N))))) + ((double) log(((double) sqrt((((double) (N + 1.0)) / N)))))));
	} else {
		VAR = ((double) ((1.0 / N) * ((double) (1.0 + (((double) ((0.3333333333333333 / N) - 0.5)) / N)))));
	}
	return VAR;
}

Error

Bits error versus N

Derivation

1. Split input into 2 regimes

2. if N < 7318.37632317876796

   1. Initial program 0.1

      \[\log \left(N + 1\right) - \log N\]
   2. Using strategy rm

   3. Applied diff-log 0.0

      \[\leadsto \color{blue}{\log \left(\frac{N + 1}{N}\right)}\]
   4. Using strategy rm

   5. Applied add-sqr-sqrt 0.1

      \[\leadsto \log \color{blue}{\left(\sqrt{\frac{N + 1}{N}} \cdot \sqrt{\frac{N + 1}{N}}\right)}\]

   6. Applied log-prod 0.1

      \[\leadsto \color{blue}{\log \left(\sqrt{\frac{N + 1}{N}}\right) + \log \left(\sqrt{\frac{N + 1}{N}}\right)}\]

   if 7318.37632317876796 < N

   1. Initial program 59.6

      \[\log \left(N + 1\right) - \log N\]

   2. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{\left(0.333333333333333315 \cdot \frac{1}{{N}^{3}} + 1 \cdot \frac{1}{N}\right) - 0.5 \cdot \frac{1}{{N}^{2}}}\]

   3. Simplified 0.0

      \[\leadsto \color{blue}{\frac{1}{N} \cdot \left(1 + \frac{\frac{0.333333333333333315}{N} - 0.5}{N}\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 7318.37632317876796:\\ \;\;\;\;\log \left(\sqrt{\frac{N + 1}{N}}\right) + \log \left(\sqrt{\frac{N + 1}{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} \cdot \left(1 + \frac{\frac{0.333333333333333315}{N} - 0.5}{N}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020181 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1.0)) (log N)))